Dividing Rotated Numbers
Define "Rotation" of a (multidigit) number to involve taking the first
digit of the number and putting it at the end to form a new number.
For example, successive rotations of 1234 yield the numbers
2341, 3412 and 4123
(Another rotation would give you back the original number).

Can there exist a number, consisting of only distinct digits,
all of whose "rotations" (including the number itself) are exact
multiples of each of its digits?

Can there exist a number, consisting of at least 2 distinct digits
(repetition of digits allowed), all of whose "rotations"
(including the number itself) are exact multiples of its distinct digits?
Please logically explain your results.
Source: Reader Sudipta Das.
Solution
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